Practical activity

Duration2h30

Presentation & objectives

Here are a few exercises to make you manipulate simple programs and your IDE.

During these activity, please pay attention to the good programming practices you studied earlier. These are good reflexes to acquire quickly.

For each exercise, we ask you to show your solution to another student, so that they can give you remarks on readability.

Important

The aim of this session is to help you master the basics of algorithms and programming by tackling very classic and simple problems. An intelligent programming assistant such as GitHub Copilot, that you may have installed already, will be able to provide you with a solution to these exercises based only on a wisely chosen file name.

For the sake of training, we advise you to postpone the discovery of Copilot’s functionalities for now.

Important

We provide you the solutions to the exercises. Make sure to check them only after you have a solution to the exercises, for comparison purpose! Even if you are sure your solution is correct, please have a look at them, as they sometimes provide additional elements you may have missed.

Activity contents

1 — Fixing syntactic bugs

The following code contains a syntax error. Use the Error tab of your IDE find the location of the error. What does it say?

​
# The string to manipulate
s = "IMT Atlantique"
print(s)

# Loop to append to a new string
result = ""
for i j in range(len(s)):
    result = s[i] + result

# Print the result
print("Reversed string:", result)
/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // The string to manipulate
        String s = "IMT Atlantique";
        System.out.println(s);

        // Loop to append to a new string
        String result = "";
        for (int i j = 0; i < s.length(); i++) {
            result = s.charAt(i) + result;
        }

        // Print the result
        System.out.println("Reversed string: " + result);
    }

}
Correction
Output
Expected "in" Pylance [Ln 7, Col 7]
Unexpected indentation Pylance [Ln 8, Col 1]
Unindent not expected Pylance [Ln 11, Col 1]
Main.java:25: error: ';' expected
        for (int i j = 0; i < s.length(); i++) {
                  ^

As you can see, the Problems tab does not directly tell you that there is an extra j. As it is trying to parse the program, it encounters something unexpected, and indicates it to you. In this case, the shown errors are indirect consequences of your syntax error.

Therefore, you may sometimes need some interpretation to link error messages to the actual errors to correct. With time and experience, you will learn to interprete these messages efficiently.

Check again your code. Your IDE should have underlined the lines that correspond to the problems identified above. This will help you find errors more easily.

Try to fix the error and run the code again to see the difference!

Correction
​
# The string to manipulate
s = "IMT Atlantique"
print(s)

# Loop to append to a new string
result = ""
for i in range(len(s)):
    result = s[i] + result

# Print the result
print("Reversed string:", result)
/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // The string to manipulate
        String s = "IMT Atlantique";
        System.out.println(s);

        // Loop to append to a new string
        String result = "";
        for (int i = 0; i < s.length(); i++) {
            result = s.charAt(i) + result;
        }

        // Print the result
        System.out.println("Reversed string: " + result);
    }

}

2 — Fixing semantic bugs

Here is a code that is syntactically correct:

​
# Needed imports
import random

# Fill a list with 10 random numbers
numbers = []
for i in range(10):
    numbers.append(random.randint(-50, 50))

# Let's do the following operation 50 times
for i in range(50):

    # Divide each number by the subtraction of the two next numbers in the list
    for j in range(len(numbers) - 2):
        numbers[j] /= numbers[j + 1] - numbers[j + 2]

# Print the final list
print("Final list", numbers)
// Needed imports
import java.util.ArrayList;
import java.util.List;
import java.util.Random;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Use a random number generator
        Random random = new Random();

        // Fill a list with 10 random numbers
        List<Double> numbers = new ArrayList<>();
        for (int i = 0; i < 10; i++) {
            numbers.add((double) (random.nextInt(101) - 50));
        }

        // Let's do the following operation 50 times
        for (int k = 0; k < 50; k++) {
            // Divide each number by the subtraction of the two next numbers in the list
            for (int j = 0; j < numbers.size() - 2; j++) {
                numbers.set(j, numbers.get(j) / (numbers.get(j + 1) - numbers.get(j + 2)));
            }
        }

        // Print the final list
        System.out.println("Final list " + numbers);
    }

}

However, when running it, we get the following output:

Output
Traceback (most recent call last):
  File "session_1.py", line 14, in <module>
    numbers[j] /= numbers[j + 1] - numbers[j + 2]
ZeroDivisionError: float division by zero
Final list [0.0, 0.0, -Infinity, 0.0, 0.0, Infinity, 5.139385345176833E-80, 2.240361480978435E-62, -41.0, -23.0]
# Here, no error is raised, but the values `Infinity` and `-Infinity` indicate that the division by zero was not handled correctly

Use the print command to find the source of the error. I.e., add some print where it pertinent to understand what leads to the error. Then run the code and try to analyze why it happens.

Correction

Before anything, note that the program uses random numbers. Therefore, we should set the seed to a value that leads to the error. On our computer, seed 42 leads to an error. It may be different for you, try to find such a value.

​
# Needed imports
import random

# Set the seed
random.seed(42)

# Fill a list with 10 random numbers
numbers = []
for i in range(10):
    numbers.append(random.randint(-50, 50))

# Let's do the following operation 50 times
for i in range(50):

    # Divide each number by the subtraction of the two next numbers in the list
    for j in range(len(numbers) - 2):
        numbers[j] /= numbers[j + 1] - numbers[j + 2]

# Print the final list
print("Final list", numbers)
// Needed imports
import java.util.ArrayList;
import java.util.List;
import java.util.Random;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Use a random number generator
        Random random = new Random();

        // Set the seed
        random.setSeed(42);

        // Fill a list with 10 random numbers
        List<Double> numbers = new ArrayList<>();
        for (int i = 0; i < 10; i++) {
            numbers.add((double) (random.nextInt(101) - 50));
        }

        // Let's do the following operation 50 times
        for (int k = 0; k < 50; k++) {
            // Divide each number by the subtraction of the two next numbers in the list
            for (int j = 0; j < numbers.size() - 2; j++) {
                numbers.set(j, numbers.get(j) / (numbers.get(j + 1) - numbers.get(j + 2)));
            }
        }

        // Print the final list
        System.out.println("Final list " + numbers);
    }

}

At this point, we get the following output when running the program:

Output
Traceback (most recent call last):
  File "session_1.py", line 17, in <module>
    numbers[j] /= numbers[j + 1] - numbers[j + 2]
ZeroDivisionError: float division by zero
Final list [-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -2.8530960909304215E-69, -5.679991152181034E-82, 25.0, -20.0]
# Here, no error is raised, but the values `Infinity` and `-Infinity` indicate that the division by zero was not handled correctly

The error message tells us that the crash occurs in the last loop. It seems to be related to the subtraction of two numbers being 0. Let’s have a look at the numbers list before that loop, and what parts of it lead to a problem.

​
# Needed imports
import random

# Set the seed
random.seed(42)

# Fill a list with 10 random numbers
numbers = []
for i in range(10):
    numbers.append(random.randint(-50, 50))

# Let's do the following operation 50 times
for i in range(50):

    # Debug
    print("Current list", numbers)

    # Divide each number by the subtraction of the two next numbers in the list
    for j in range(len(numbers) - 2):

        # Debug
        print("  numbers[j+1] =", numbers[j + 1], "/ numbers[j+2] =", numbers[j + 2], "/ difference =", numbers[j + 1] - numbers[j + 2])

        numbers[j] /= numbers[j + 1] - numbers[j + 2]

# Print the final list
print("Final list", numbers)
// Needed imports
import java.util.ArrayList;
import java.util.List;
import java.util.Random;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Use a random number generator
        Random random = new Random();

        // Set the seed
        random.setSeed(42);

        // Fill a list with 10 random numbers
        List<Double> numbers = new ArrayList<>();
        for (int i = 0; i < 10; i++) {
            numbers.add((double) (random.nextInt(101) - 50));
        }

        // Let's do the following operation 50 times
        for (int k = 0; k < 50; k++) {
            // Debug
            System.out.println(numbers);

            // Divide each number by the subtraction of the two next numbers in the list
            for (int j = 0; j < numbers.size() - 2; j++) {
                // Debug
                System.out.println("  numbers[j+1] = " + numbers.get(j + 1) + "/ numbers[j+2] = " + numbers.get(j + 2) + "/ division = " + (numbers.get(j + 1) / numbers.get(j + 2)));

                numbers.set(j, numbers.get(j) / (numbers.get(j + 1) - numbers.get(j + 2)));
            }
        }

        // Print the final list
        System.out.println("Final list " + numbers);
    }

}

Here is what we get:

Output
Current list [31, -36, -47, 44, -15, -19, -22, -33, 44, -37]
  numbers[j+1] = -36 / numbers[j+2] = -47 / difference = 11
  numbers[j+1] = -47 / numbers[j+2] = 44 / difference = -91
  numbers[j+1] = 44 / numbers[j+2] = -15 / difference = 59
  numbers[j+1] = -15 / numbers[j+2] = -19 / difference = 4
  numbers[j+1] = -19 / numbers[j+2] = -22 / difference = 3
  numbers[j+1] = -22 / numbers[j+2] = -33 / difference = 11
  numbers[j+1] = -33 / numbers[j+2] = 44 / difference = -77
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [2.8181818181818183, 0.3956043956043956, -0.7966101694915254, 11.0, -5.0, -1.7272727272727273, 0.2857142857142857, -0.4074074074074074, 44, -37]
  numbers[j+1] = 0.3956043956043956 / numbers[j+2] = -0.7966101694915254 / difference = 1.192214565095921
  numbers[j+1] = -0.7966101694915254 / numbers[j+2] = 11.0 / difference = -11.796610169491526
  numbers[j+1] = 11.0 / numbers[j+2] = -5.0 / difference = 16.0
  numbers[j+1] = -5.0 / numbers[j+2] = -1.7272727272727273 / difference = -3.2727272727272725
  numbers[j+1] = -1.7272727272727273 / numbers[j+2] = 0.2857142857142857 / difference = -2.012987012987013
  numbers[j+1] = 0.2857142857142857 / numbers[j+2] = -0.4074074074074074 / difference = 0.693121693121693
  numbers[j+1] = -0.4074074074074074 / numbers[j+2] = 44 / difference = -44.407407407407405
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [2.3638209938787975, -0.03353543008715422, -0.04978813559322034, -3.361111111111111, 2.4838709677419355, -2.492019430950729, -0.006433933039437627, -0.005029721079103795, 44, -37]
  numbers[j+1] = -0.03353543008715422 / numbers[j+2] = -0.04978813559322034 / difference = 0.01625270550606612
  numbers[j+1] = -0.04978813559322034 / numbers[j+2] = -3.361111111111111 / difference = 3.311322975517891
  numbers[j+1] = -3.361111111111111 / numbers[j+2] = 2.4838709677419355 / difference = -5.844982078853047
  numbers[j+1] = 2.4838709677419355 / numbers[j+2] = -2.492019430950729 / difference = 4.975890398692664
  numbers[j+1] = -2.492019430950729 / numbers[j+2] = -0.006433933039437627 / difference = -2.4855854979112912
  numbers[j+1] = -0.006433933039437627 / numbers[j+2] = -0.005029721079103795 / difference = -0.0014042119603338322
  numbers[j+1] = -0.005029721079103795 / numbers[j+2] = 44 / difference = -44.005029721079104
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [145.4416923383329, -0.010127502009044973, 0.008518098930252016, -0.6754793296882482, -0.9993102107447938, 1774.6746939531017, 0.0001462090374718159, -6.209532196424438e-05, 44, -37]
  numbers[j+1] = -0.010127502009044973 / numbers[j+2] = 0.008518098930252016 / difference = -0.01864560093929699
  numbers[j+1] = 0.008518098930252016 / numbers[j+2] = -0.6754793296882482 / difference = 0.6839974286185002
  numbers[j+1] = -0.6754793296882482 / numbers[j+2] = -0.9993102107447938 / difference = 0.32383088105654556
  numbers[j+1] = -0.9993102107447938 / numbers[j+2] = 1774.6746939531017 / difference = -1775.6740041638466
  numbers[j+1] = 1774.6746939531017 / numbers[j+2] = 0.0001462090374718159 / difference = 1774.6745477440643
  numbers[j+1] = 0.0001462090374718159 / numbers[j+2] = -6.209532196424438e-05 / difference = 0.00020830435943606026
  numbers[j+1] = -6.209532196424438e-05 / numbers[j+2] = 44 / difference = -44.000062095321965
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [-7800.32206051368, -0.014806345148840596, 0.026304158832723037, 0.0003804072865313625, -0.0005630949133829072, 8519623.395101551, -3.32292798030757e-06, -7.666089131388194e-07, 44, -37]
  numbers[j+1] = -0.014806345148840596 / numbers[j+2] = 0.026304158832723037 / difference = -0.041110503981563636
  numbers[j+1] = 0.026304158832723037 / numbers[j+2] = 0.0003804072865313625 / difference = 0.025923751546191674
  numbers[j+1] = 0.0003804072865313625 / numbers[j+2] = -0.0005630949133829072 / difference = 0.0009435021999142697
  numbers[j+1] = -0.0005630949133829072 / numbers[j+2] = 8519623.395101551 / difference = -8519623.395664645
  numbers[j+1] = 8519623.395101551 / numbers[j+2] = -3.32292798030757e-06 / difference = 8519623.395104874
  numbers[j+1] = -3.32292798030757e-06 / numbers[j+2] = -7.666089131388194e-07 / difference = -2.5563190671687508e-06
  numbers[j+1] = -7.666089131388194e-07 / numbers[j+2] = 44 / difference = -44.00000076660891
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [189740.36572287712, -0.5711497860353364, 27.879276630317488, -4.465071621885766e-11, -6.609387378630434e-11, -3332769959947.7046, 7.552108914573704e-08, -9.464307569615054e-09, 44, -37]
  numbers[j+1] = -0.5711497860353364 / numbers[j+2] = 27.879276630317488 / difference = -28.450426416352823
  numbers[j+1] = 27.879276630317488 / numbers[j+2] = -4.465071621885766e-11 / difference = 27.87927663036214
  numbers[j+1] = -4.465071621885766e-11 / numbers[j+2] = -6.609387378630434e-11 / difference = 2.144315756744668e-11
  numbers[j+1] = -6.609387378630434e-11 / numbers[j+2] = -3332769959947.7046 / difference = 3332769959947.7046
  numbers[j+1] = -3332769959947.7046 / numbers[j+2] = 7.552108914573704e-08 / difference = -3332769959947.7046
  numbers[j+1] = 7.552108914573704e-08 / numbers[j+2] = -9.464307569615054e-09 / difference = 8.49853967153521e-08
  numbers[j+1] = -9.464307569615054e-09 / numbers[j+2] = 44 / difference = -44.00000000946431
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [-6669.1571840138595, -0.020486535343364014, 1300147916305.0884, -1.3397479200622142e-23, 1.9831513900029704e-23, -3.9215795757362864e+19, -1.7163883893066504e-09, -1.168433033285809e-10, 44, -37]
  numbers[j+1] = -0.020486535343364014 / numbers[j+2] = 1300147916305.0884 / difference = -1300147916305.109
  numbers[j+1] = 1300147916305.0884 / numbers[j+2] = -1.3397479200622142e-23 / difference = 1300147916305.0884
  numbers[j+1] = -1.3397479200622142e-23 / numbers[j+2] = 1.9831513900029704e-23 / difference = -3.3228993100651846e-23
  numbers[j+1] = 1.9831513900029704e-23 / numbers[j+2] = -3.9215795757362864e+19 / difference = 3.9215795757362864e+19
  numbers[j+1] = -3.9215795757362864e+19 / numbers[j+2] = -1.7163883893066504e-09 / difference = -3.9215795757362864e+19
  numbers[j+1] = -1.7163883893066504e-09 / numbers[j+2] = -1.168433033285809e-10 / difference = -1.5995450859780694e-09
  numbers[j+1] = -1.168433033285809e-10 / numbers[j+2] = 44 / difference = -44.00000000011684
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [5.129537262934622e-09, -1.5757080472493495e-14, -3.9126912824800088e+34, -3.416347658355685e-43, -5.057021926249269e-43, 2.451684300813796e+28, 3.900882702959301e-11, -1.4425099176368013e-12, 44, -37]
  numbers[j+1] = -1.5757080472493495e-14 / numbers[j+2] = -3.9126912824800088e+34 / difference = 3.9126912824800088e+34
  numbers[j+1] = -3.9126912824800088e+34 / numbers[j+2] = -3.416347658355685e-43 / difference = -3.9126912824800088e+34
  numbers[j+1] = -3.416347658355685e-43 / numbers[j+2] = -5.057021926249269e-43 / difference = 1.640674267893584e-43
  numbers[j+1] = -5.057021926249269e-43 / numbers[j+2] = 2.451684300813796e+28 / difference = -2.451684300813796e+28
  numbers[j+1] = 2.451684300813796e+28 / numbers[j+2] = 3.900882702959301e-11 / difference = 2.451684300813796e+28
  numbers[j+1] = 3.900882702959301e-11 / numbers[j+2] = -1.4425099176368013e-12 / difference = 4.0451336947229815e-11
  numbers[j+1] = -1.4425099176368013e-12 / numbers[j+2] = 44 / difference = -44.00000000000144
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [1.3109997422754321e-43, 4.027171922060274e-49, -2.3848068803464593e+77, 1.3934696474671249e-71, -2.0626725572173683e-71, 6.060823908025844e+38, -8.865642506725394e-13, -1.780876441526915e-14, 44, -37]
  numbers[j+1] = 4.027171922060274e-49 / numbers[j+2] = -2.3848068803464593e+77 / difference = 2.3848068803464593e+77
  numbers[j+1] = -2.3848068803464593e+77 / numbers[j+2] = 1.3934696474671249e-71 / difference = -2.3848068803464593e+77
  numbers[j+1] = 1.3934696474671249e-71 / numbers[j+2] = -2.0626725572173683e-71 / difference = 3.456142204684493e-71
  numbers[j+1] = -2.0626725572173683e-71 / numbers[j+2] = 6.060823908025844e+38 / difference = -6.060823908025844e+38
  numbers[j+1] = 6.060823908025844e+38 / numbers[j+2] = -8.865642506725394e-13 / difference = 6.060823908025844e+38
  numbers[j+1] = -8.865642506725394e-13 / numbers[j+2] = -1.780876441526915e-14 / difference = -8.687554862572702e-13
  numbers[j+1] = -1.780876441526915e-14 / numbers[j+2] = 44 / difference = -44.00000000000002
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [5.497299395936719e-121, -1.6886784230827175e-126, -6.900198947584002e+147, -2.2991422760556847e-110, -3.403287388841545e-110, -6.9764438946299935e+50, 2.0149187515284978e-14, -2.1986128907739692e-16, 44, -37]
  numbers[j+1] = -1.6886784230827175e-126 / numbers[j+2] = -6.900198947584002e+147 / difference = 6.900198947584002e+147
  numbers[j+1] = -6.900198947584002e+147 / numbers[j+2] = -2.2991422760556847e-110 / difference = -6.900198947584002e+147
  numbers[j+1] = -2.2991422760556847e-110 / numbers[j+2] = -3.403287388841545e-110 / difference = 1.1041451127858606e-110
  numbers[j+1] = -3.403287388841545e-110 / numbers[j+2] = -6.9764438946299935e+50 / difference = 6.9764438946299935e+50
  numbers[j+1] = -6.9764438946299935e+50 / numbers[j+2] = 2.0149187515284978e-14 / difference = -6.9764438946299935e+50
  numbers[j+1] = 2.0149187515284978e-14 / numbers[j+2] = -2.1986128907739692e-16 / difference = 2.0369048804362376e-14
  numbers[j+1] = -2.1986128907739692e-16 / numbers[j+2] = 44 / difference = -44.0
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [7.96687086516761e-269, 2.4472894707970443e-274, -6.249358773299426e+257, -3.295579109903561e-161, 4.8782552260775314e-161, -3.42502193481704e+64, -4.579360798928404e-16, -2.7143369021900854e-18, 44, -37]
  numbers[j+1] = 2.4472894707970443e-274 / numbers[j+2] = -6.249358773299426e+257 / difference = 6.249358773299426e+257
  numbers[j+1] = -6.249358773299426e+257 / numbers[j+2] = -3.295579109903561e-161 / difference = -6.249358773299426e+257
  numbers[j+1] = -3.295579109903561e-161 / numbers[j+2] = 4.8782552260775314e-161 / difference = -8.173834335981092e-161
  numbers[j+1] = 4.8782552260775314e-161 / numbers[j+2] = -3.42502193481704e+64 / difference = 3.42502193481704e+64
  numbers[j+1] = -3.42502193481704e+64 / numbers[j+2] = -4.579360798928404e-16 / difference = -3.42502193481704e+64
  numbers[j+1] = -4.579360798928404e-16 / numbers[j+2] = -2.7143369021900854e-18 / difference = -4.552217429906503e-16
  numbers[j+1] = -2.7143369021900854e-18 / numbers[j+2] = 44 / difference = -44.0
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [0.0, -0.0, inf, -9.622067165183299e-226, -1.424298973529967e-225, 7.523854006436124e+79, 1.0407638179382736e-17, -3.351033212580352e-20, 44, -37]
  numbers[j+1] = -0.0 / numbers[j+2] = inf / difference = -inf
  numbers[j+1] = inf / numbers[j+2] = -9.622067165183299e-226 / difference = inf
  numbers[j+1] = -9.622067165183299e-226 / numbers[j+2] = -1.424298973529967e-225 / difference = 4.620922570116372e-226
  numbers[j+1] = -1.424298973529967e-225 / numbers[j+2] = 7.523854006436124e+79 / difference = -7.523854006436124e+79
  numbers[j+1] = 7.523854006436124e+79 / numbers[j+2] = 1.0407638179382736e-17 / difference = 7.523854006436124e+79
  numbers[j+1] = 1.0407638179382736e-17 / numbers[j+2] = -3.351033212580352e-20 / difference = 1.044114851150854e-17
  numbers[j+1] = -3.351033212580352e-20 / numbers[j+2] = 44 / difference = -44.0
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [-0.0, -0.0, inf, 1.2788747837148755e-305, -1.8930444055820065e-305, 7.205963978141975e+96, -2.3653723134960764e-19, -4.137078040222657e-22, 44, -37]
  numbers[j+1] = -0.0 / numbers[j+2] = inf / difference = -inf
  numbers[j+1] = inf / numbers[j+2] = 1.2788747837148755e-305 / difference = inf
  numbers[j+1] = 1.2788747837148755e-305 / numbers[j+2] = -1.8930444055820065e-305 / difference = 3.171919189296882e-305
  numbers[j+1] = -1.8930444055820065e-305 / numbers[j+2] = 7.205963978141975e+96 / difference = -7.205963978141975e+96
  numbers[j+1] = 7.205963978141975e+96 / numbers[j+2] = -2.3653723134960764e-19 / difference = 7.205963978141975e+96
  numbers[j+1] = -2.3653723134960764e-19 / numbers[j+2] = -4.137078040222657e-22 / difference = -2.3612352354558536e-19
  numbers[j+1] = -4.137078040222657e-22 / numbers[j+2] = 44 / difference = -44.0
  numbers[j+1] = 44 / numbers[j+2] = -37 / difference = 81
Current list [0.0, -0.0, inf, -0.0, -0.0, -3.051777251981763e+115, 5.375846167036538e-21, -5.107503753361305e-24, 44, -37]
  numbers[j+1] = -0.0 / numbers[j+2] = inf / difference = -inf
  numbers[j+1] = inf / numbers[j+2] = -0.0 / difference = inf
  numbers[j+1] = -0.0 / numbers[j+2] = -0.0 / difference = 0.0
Traceback (most recent call last):
  File "session_1.py", line 21, in <module>
    numbers[j] /= numbers[j + 1] - numbers[j + 2]
ZeroDivisionError: float division by zero
[-35.0, 9.0, -46.0, 39.0, -39.0, 6.0, -47.0, -26.0, 25.0, -20.0]
  numbers[j+1] = 9.0/ numbers[j+2] = -46.0/ division = -0.1956521739130435
  numbers[j+1] = -46.0/ numbers[j+2] = 39.0/ division = -1.1794871794871795
  numbers[j+1] = 39.0/ numbers[j+2] = -39.0/ division = -1.0
  numbers[j+1] = -39.0/ numbers[j+2] = 6.0/ division = -6.5
  numbers[j+1] = 6.0/ numbers[j+2] = -47.0/ division = -0.1276595744680851
  numbers[j+1] = -47.0/ numbers[j+2] = -26.0/ division = 1.8076923076923077
  numbers[j+1] = -26.0/ numbers[j+2] = 25.0/ division = -1.04
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.6363636363636364, -0.10588235294117647, -0.5897435897435898, -0.8666666666666667, -0.7358490566037735, -0.2857142857142857, 0.9215686274509803, -0.5777777777777777, 25.0, -20.0]
  numbers[j+1] = -0.10588235294117647/ numbers[j+2] = -0.5897435897435898/ division = 0.179539641943734
  numbers[j+1] = -0.5897435897435898/ numbers[j+2] = -0.8666666666666667/ division = 0.6804733727810651
  numbers[j+1] = -0.8666666666666667/ numbers[j+2] = -0.7358490566037735/ division = 1.1777777777777778
  numbers[j+1] = -0.7358490566037735/ numbers[j+2] = -0.2857142857142857/ division = 2.5754716981132075
  numbers[j+1] = -0.2857142857142857/ numbers[j+2] = 0.9215686274509803/ division = -0.3100303951367781
  numbers[j+1] = 0.9215686274509803/ numbers[j+2] = -0.5777777777777777/ division = -1.5950226244343892
  numbers[j+1] = -0.5777777777777777/ numbers[j+2] = 25.0/ division = -0.02311111111111111
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-1.315177964180458, -0.38235294117647056, 4.508136094674554, 1.9253493013972058, 0.6095083833121744, -0.19055922281728732, -0.036030050595390196, -0.012839506172839505, 25.0, -20.0]
  numbers[j+1] = -0.38235294117647056/ numbers[j+2] = 4.508136094674554/ division = -0.08481397481059696
  numbers[j+1] = 4.508136094674554/ numbers[j+2] = 1.9253493013972058/ division = 2.341464009363416
  numbers[j+1] = 1.9253493013972058/ numbers[j+2] = 0.6095083833121744/ division = 3.1588561439213083
  numbers[j+1] = 0.6095083833121744/ numbers[j+2] = -0.19055922281728732/ division = -3.1985247121656526
  numbers[j+1] = -0.19055922281728732/ numbers[j+2] = -0.036030050595390196/ division = 5.288896897682073
  numbers[j+1] = -0.036030050595390196/ numbers[j+2] = -0.012839506172839505/ division = 2.8061866329101983
  numbers[j+1] = -0.012839506172839505/ numbers[j+2] = 25.0/ division = -5.135802469135802E-4
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.26892565437509375, -0.1480389098208511, 3.4260494811449784, 2.4064832604729385, -3.9442933301742342, 8.217108634672064, 0.0014404622308674094, -2.853223593964335E-4, 25.0, -20.0]
  numbers[j+1] = -0.1480389098208511/ numbers[j+2] = 3.4260494811449784/ division = -0.04320979910989984
  numbers[j+1] = 3.4260494811449784/ numbers[j+2] = 2.4064832604729385/ division = 1.4236747611831166
  numbers[j+1] = 2.4064832604729385/ numbers[j+2] = -3.9442933301742342/ division = -0.6101177217381637
  numbers[j+1] = -3.9442933301742342/ numbers[j+2] = 8.217108634672064/ division = -0.48000988006064577
  numbers[j+1] = 8.217108634672064/ numbers[j+2] = 0.0014404622308674094/ division = 5704.494334241538
  numbers[j+1] = 0.0014404622308674094/ numbers[j+2] = -2.853223593964335E-4/ division = -5.0485431072227955
  numbers[j+1] = -2.853223593964335E-4/ numbers[j+2] = 25.0/ division = -1.1412894375857339E-5
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.07524314593193977, -0.14519793498383293, 0.5394693754761493, -0.197878769851462, -0.48009404072635886, 4761.375597527968, -5.761783164846961E-5, -6.3404968754763E-6, 25.0, -20.0]
  numbers[j+1] = -0.14519793498383293/ numbers[j+2] = 0.5394693754761493/ division = -0.2691495413538119
  numbers[j+1] = 0.5394693754761493/ numbers[j+2] = -0.197878769851462/ division = -2.7262620233646224
  numbers[j+1] = -0.197878769851462/ numbers[j+2] = -0.48009404072635886/ division = 0.4121666862435557
  numbers[j+1] = -0.48009404072635886/ numbers[j+2] = 4761.375597527968/ division = -1.0083095334373877E-4
  numbers[j+1] = 4761.375597527968/ numbers[j+2] = -5.761783164846961E-5/ division = -8.263718819856761E7
  numbers[j+1] = -5.761783164846961E-5/ numbers[j+2] = -6.3404968754763E-6/ division = 9.087273880904064
  numbers[j+1] = -6.3404968754763E-6/ numbers[j+2] = 25.0/ division = -2.53619875019052E-7
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.10989738341880077, -0.1969191024672884, 1.9115527441294646, 4.155496988323788E-5, -1.008309521235744E-4, -9.285536423854238E7, 2.304712681417842E-6, -1.4089993056614E-7, 25.0, -20.0]
  numbers[j+1] = -0.1969191024672884/ numbers[j+2] = 1.9115527441294646/ division = -0.10301525975259805
  numbers[j+1] = 1.9115527441294646/ numbers[j+2] = 4.155496988323788E-5/ division = 46000.5806646134
  numbers[j+1] = 4.155496988323788E-5/ numbers[j+2] = -1.008309521235744E-4/ division = -0.4121251362608355
  numbers[j+1] = -1.008309521235744E-4/ numbers[j+2] = -9.285536423854238E7/ division = 1.0858925916713115E-12
  numbers[j+1] = -9.285536423854238E7/ numbers[j+2] = 2.304712681417842E-6/ division = -4.028934495271595E13
  numbers[j+1] = 2.304712681417842E-6/ numbers[j+2] = -1.4089993056614E-7/ division = -16.357088837144488
  numbers[j+1] = -1.4089993056614E-7/ numbers[j+2] = 25.0/ division = -5.6359972226455995E-9
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.05212181684862627, -0.10301749923518168, 13425.152691977572, 4.4752363230765706E-13, 1.0858925916712844E-12, -3.796814089996627E13, -9.218850673713952E-8, -3.1311095681364443E-9, 25.0, -20.0]
  numbers[j+1] = -0.10301749923518168/ numbers[j+2] = 13425.152691977572/ division = -7.673469464279654E-6
  numbers[j+1] = 13425.152691977572/ numbers[j+2] = 4.4752363230765706E-13/ division = 2.9998756987984584E16
  numbers[j+1] = 4.4752363230765706E-13/ numbers[j+2] = 1.0858925916712844E-12/ division = 0.4121251362612933
  numbers[j+1] = 1.0858925916712844E-12/ numbers[j+2] = -3.796814089996627E13/ division = -2.860009908128657E-26
  numbers[j+1] = -3.796814089996627E13/ numbers[j+2] = -9.218850673713952E-8/ division = 4.118533019330298E20
  numbers[j+1] = -9.218850673713952E-8/ numbers[j+2] = -3.1311095681364443E-9/ division = 29.442759740920767
  numbers[j+1] = -3.1311095681364443E-9/ numbers[j+2] = 25.0/ division = -1.2524438272545777E-10
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[3.882370509474452E-6, -7.673469464279654E-6, -2.103039706905665E16, 1.1786819730961717E-26, -2.860009908128657E-26, 4.263333772029585E20, 3.687540269023737E-9, -6.958021262525432E-11, 25.0, -20.0]
  numbers[j+1] = -7.673469464279654E-6/ numbers[j+2] = -2.103039706905665E16/ division = 3.6487515851852907E-22
  numbers[j+1] = -2.103039706905665E16/ numbers[j+2] = 1.1786819730961717E-26/ division = -1.7842299745888047E42
  numbers[j+1] = 1.1786819730961717E-26/ numbers[j+2] = -2.860009908128657E-26/ division = -0.4121251362612933
  numbers[j+1] = -2.860009908128657E-26/ numbers[j+2] = 4.263333772029585E20/ division = -6.70838846090892E-47
  numbers[j+1] = 4.263333772029585E20/ numbers[j+2] = 3.687540269023737E-9/ division = 1.1561456854702463E29
  numbers[j+1] = 3.687540269023737E-9/ numbers[j+2] = -6.958021262525432E-11/ division = -52.99696752701981
  numbers[j+1] = -6.958021262525432E-11/ numbers[j+2] = 25.0/ division = -2.783208505010173E-12
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[1.8460757049551046E-22, 3.6487515851852907E-22, -5.2072298871878E41, -2.7646955085457766E-47, -6.70838846090892E-47, 1.1347343777909453E29, -1.4750161076053896E-10, -1.5462269472278738E-12, 25.0, -20.0]
  numbers[j+1] = 3.6487515851852907E-22/ numbers[j+2] = -5.2072298871878E41/ division = -7.007087576761132E-64
  numbers[j+1] = -5.2072298871878E41/ numbers[j+2] = -2.7646955085457766E-47/ division = 1.8834731966294512E88
  numbers[j+1] = -2.7646955085457766E-47/ numbers[j+2] = -6.70838846090892E-47/ division = 0.4121251362612933
  numbers[j+1] = -6.70838846090892E-47/ numbers[j+2] = 1.1347343777909453E29/ division = -5.911857957426599E-76
  numbers[j+1] = 1.1347343777909453E29/ numbers[j+2] = -1.4750161076053896E-10/ division = -7.693030414651719E38
  numbers[j+1] = -1.4750161076053896E-10/ numbers[j+2] = -1.5462269472278738E-12/ division = 95.39454154837016
  numbers[j+1] = -1.5462269472278738E-12/ numbers[j+2] = 25.0/ division = -6.184907788911496E-14
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[3.5452164489555316E-64, -7.007087576761132E-64, -1.3203943486694417E88, 2.4364252662618483E-76, -5.911857957426599E-76, -7.77452909337256E38, 5.9000644304211934E-12, -3.4360598827286084E-14, 25.0, -20.0]
  numbers[j+1] = -7.007087576761132E-64/ numbers[j+2] = -1.3203943486694417E88/ division = 5.3068142739494135E-152
  numbers[j+1] = -1.3203943486694417E88/ numbers[j+2] = 2.4364252662618483E-76/ division = -5.419391954899945E163
  numbers[j+1] = 2.4364252662618483E-76/ numbers[j+2] = -5.911857957426599E-76/ division = -0.4121251362612933
  numbers[j+1] = -5.911857957426599E-76/ numbers[j+2] = -7.77452909337256E38/ division = 7.60413638745811E-115
  numbers[j+1] = -7.77452909337256E38/ numbers[j+2] = 5.9000644304211934E-12/ division = -1.3177024056358571E50
  numbers[j+1] = 5.9000644304211934E-12/ numbers[j+2] = -3.4360598827286084E-14/ division = -171.71017478705568
  numbers[j+1] = -3.4360598827286084E-14/ numbers[j+2] = 25.0/ division = -1.3744239530914434E-15
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[2.6849679056321607E-152, 5.3068142739494135E-152, -1.581635784615922E163, 3.1338557448306322E-115, 7.60413638745811E-115, -1.3100728469994853E50, -2.360025772168474E-13, -7.635688628285796E-16, 25.0, -20.0]
  numbers[j+1] = 5.3068142739494135E-152/ numbers[j+2] = -1.581635784615922E163/ division = -3.35526948E-315
  numbers[j+1] = -1.581635784615922E163/ numbers[j+2] = 3.1338557448306322E-115/ division = -5.04693232043264E277
  numbers[j+1] = 3.1338557448306322E-115/ numbers[j+2] = 7.60413638745811E-115/ division = 0.4121251362612933
  numbers[j+1] = 7.60413638745811E-115/ numbers[j+2] = -1.3100728469994853E50/ division = -5.8043614939995E-165
  numbers[j+1] = -1.3100728469994853E50/ numbers[j+2] = -2.360025772168474E-13/ division = 5.551095511112765E62
  numbers[j+1] = -2.360025772168474E-13/ numbers[j+2] = -7.635688628285796E-16/ division = 309.0783146166998
  numbers[j+1] = -7.635688628285796E-16/ numbers[j+2] = 25.0/ division = -3.0542754513143186E-17
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[1.69758925E-315, -3.35526948E-315, 3.5381129532088862E277, 2.392123271624348E-165, -5.8043614939995E-165, 5.569113966965522E62, 9.440103088673896E-15, -1.6968196951746212E-17, 25.0, -20.0]
  numbers[j+1] = -3.35526948E-315/ numbers[j+2] = 3.5381129532088862E277/ division = -0.0
  numbers[j+1] = 3.5381129532088862E277/ numbers[j+2] = 2.392123271624348E-165/ division = Infinity
  numbers[j+1] = 2.392123271624348E-165/ numbers[j+2] = -5.8043614939995E-165/ division = -0.4121251362612933
  numbers[j+1] = -5.8043614939995E-165/ numbers[j+2] = 5.569113966965522E62/ division = -1.0422414639796209E-227
  numbers[j+1] = 5.569113966965522E62/ numbers[j+2] = 9.440103088673896E-15/ division = 5.89942071040227E76
  numbers[j+1] = 9.440103088673896E-15/ numbers[j+2] = -1.6968196951746212E-17/ division = -556.3409663100596
  numbers[j+1] = -1.6968196951746212E-17/ numbers[j+2] = 25.0/ division = -6.787278780698485E-19
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -4.2953390535977105E-228, -1.0422414639796209E-227, 5.888835770361958E76, -3.7760412354695584E-16, -3.7707104337213807E-19, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -4.2953390535977105E-228/ division = -Infinity
  numbers[j+1] = -4.2953390535977105E-228/ numbers[j+2] = -1.0422414639796209E-227/ division = 0.41212513626129327
  numbers[j+1] = -1.0422414639796209E-227/ numbers[j+2] = 5.888835770361958E76/ division = -1.7698599597990817E-304
  numbers[j+1] = 5.888835770361958E76/ numbers[j+2] = -3.7760412354695584E-16/ division = -1.5595263407206063E92
  numbers[j+1] = -3.7760412354695584E-16/ numbers[j+2] = -3.7707104337213807E-19/ division = 1001.4137393581073
  numbers[j+1] = -3.7707104337213807E-19/ numbers[j+2] = 25.0/ division = -1.5082841734885524E-20
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 7.294037770956036E-305, -1.7698599597990817E-304, -1.5610852220907494E92, 1.5104164941878233E-17, -8.379356519380846E-21, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 7.294037770956036E-305/ division = Infinity
  numbers[j+1] = 7.294037770956036E-305/ numbers[j+2] = -1.7698599597990817E-304/ division = -0.4121251362612933
  numbers[j+1] = -1.7698599597990817E-304/ numbers[j+2] = -1.5610852220907494E92/ division = 0.0
  numbers[j+1] = -1.5610852220907494E92/ numbers[j+2] = 1.5104164941878233E-17/ division = -1.033546196097502E109
  numbers[j+1] = 1.5104164941878233E-17/ numbers[j+2] = -8.379356519380846E-21/ division = -1802.544730844593
  numbers[j+1] = -8.379356519380846E-21/ numbers[j+2] = 25.0/ division = -3.3517426077523383E-22
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -1.0329731322979623E109, -6.041665976751293E-19, -1.862079226529077E-22, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -1.0329731322979623E109/ division = -0.0
  numbers[j+1] = -1.0329731322979623E109/ numbers[j+2] = -6.041665976751293E-19/ division = 1.7097488280101998E127
  numbers[j+1] = -6.041665976751293E-19/ numbers[j+2] = -1.862079226529077E-22/ division = 3244.5805155202675
  numbers[j+1] = -1.862079226529077E-22/ numbers[j+2] = 25.0/ division = -7.448316906116307E-24
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, 1.7102759457493246E127, 2.416666390700517E-20, -4.137953836731282E-24, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = 1.7102759457493246E127/ division = -0.0
  numbers[j+1] = 1.7102759457493246E127/ numbers[j+2] = 2.416666390700517E-20/ division = 7.077004721589099E146
  numbers[j+1] = 2.416666390700517E-20/ numbers[j+2] = -4.137953836731282E-24/ division = -5840.244927936481
  numbers[j+1] = -4.137953836731282E-24/ numbers[j+2] = 25.0/ division = -1.6551815346925128E-25
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, 7.07579316398299E146, -9.666665562802069E-22, -9.19545297051396E-26, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = 7.07579316398299E146/ division = -0.0
  numbers[j+1] = 7.07579316398299E146/ numbers[j+2] = -9.666665562802069E-22/ division = -7.319786867574153E167
  numbers[j+1] = -9.666665562802069E-22/ numbers[j+2] = -9.19545297051396E-26/ division = 10512.440870285667
  numbers[j+1] = -9.19545297051396E-26/ numbers[j+2] = 25.0/ division = -3.678181188205584E-27
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -7.320483231370316E167, 3.866666225120828E-23, -2.0434339934475466E-27, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -7.320483231370316E167/ division = 0.0
  numbers[j+1] = -7.320483231370316E167/ numbers[j+2] = 3.866666225120828E-23/ division = -1.8932286380993646E190
  numbers[j+1] = 3.866666225120828E-23/ numbers[j+2] = -2.0434339934475466E-27/ division = -18922.3935665142
  numbers[j+1] = -2.0434339934475466E-27/ numbers[j+2] = 25.0/ division = -8.173735973790186E-29
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -1.893128591105603E190, -1.5466664900483312E-24, -4.540964429883437E-29, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -1.893128591105603E190/ division = -0.0
  numbers[j+1] = -1.893128591105603E190/ numbers[j+2] = -1.5466664900483312E-24/ division = 1.2240056943668866E214
  numbers[j+1] = -1.5466664900483312E-24/ numbers[j+2] = -4.540964429883437E-29/ division = 34060.30841972557
  numbers[j+1] = -4.540964429883437E-29/ numbers[j+2] = 25.0/ division = -1.8163857719533748E-30
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, 1.2240416318462765E214, 6.186665960193325E-26, -1.0091032066407637E-30, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = 1.2240416318462765E214/ division = -0.0
  numbers[j+1] = 1.2240416318462765E214/ numbers[j+2] = 6.186665960193325E-26/ division = 1.9785157946494767E239
  numbers[j+1] = 6.186665960193325E-26/ numbers[j+2] = -1.0091032066407637E-30/ division = -61308.55515550602
  numbers[j+1] = -1.0091032066407637E-30/ numbers[j+2] = 25.0/ division = -4.036412826563055E-32
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, 1.9784835237287429E239, -2.47466638407733E-27, -2.2424515703128082E-32, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = 1.9784835237287429E239/ division = -0.0
  numbers[j+1] = 1.9784835237287429E239/ numbers[j+2] = -2.47466638407733E-27/ division = -7.994950496999671E265
  numbers[j+1] = -2.47466638407733E-27/ numbers[j+2] = -2.2424515703128082E-32/ division = 110355.39927991084
  numbers[j+1] = -2.2424515703128082E-32/ numbers[j+2] = 25.0/ division = -8.969806281251233E-34
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -7.995022944954163E265, 9.89866553630932E-29, -4.983225711806241E-34, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -7.995022944954163E265/ division = 0.0
  numbers[j+1] = -7.995022944954163E265/ numbers[j+2] = 9.89866553630932E-29/ division = -8.076869468543613E293
  numbers[j+1] = 9.89866553630932E-29/ numbers[j+2] = -4.983225711806241E-34/ division = -198639.7187038395
  numbers[j+1] = -4.983225711806241E-34/ numbers[j+2] = 25.0/ division = -1.9932902847224964E-35
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -8.076828807849667E293, -3.959466214523728E-30, -1.1073834915124979E-35, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -8.076828807849667E293/ division = -0.0
  numbers[j+1] = -8.076828807849667E293/ numbers[j+2] = -3.959466214523728E-30/ division = Infinity
  numbers[j+1] = -3.959466214523728E-30/ numbers[j+2] = -1.1073834915124979E-35/ division = 357551.4936669111
  numbers[j+1] = -1.1073834915124979E-35/ numbers[j+2] = 25.0/ division = -4.429533966049992E-37
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, Infinity, 1.5837864858094913E-31, -2.4608522033611066E-37, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 1.5837864858094913E-31/ division = Infinity
  numbers[j+1] = 1.5837864858094913E-31/ numbers[j+2] = -2.4608522033611066E-37/ division = -643592.6886004399
  numbers[j+1] = -2.4608522033611066E-37/ numbers[j+2] = 25.0/ division = -9.843408813444427E-39
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, Infinity, -6.335145943237966E-33, -5.4685604519135706E-39, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -6.335145943237966E-33/ division = -Infinity
  numbers[j+1] = -6.335145943237966E-33/ numbers[j+2] = -5.4685604519135706E-39/ division = 1158466.839480792
  numbers[j+1] = -5.4685604519135706E-39/ numbers[j+2] = 25.0/ division = -2.1874241807654282E-40
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -Infinity, 2.5340583772951862E-34, -1.2152356559807935E-40, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -Infinity/ division = 0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = 2.5340583772951862E-34/ division = -Infinity
  numbers[j+1] = 2.5340583772951862E-34/ numbers[j+2] = -1.2152356559807935E-40/ division = -2085240.3110654254
  numbers[j+1] = -1.2152356559807935E-40/ numbers[j+2] = 25.0/ division = -4.860942623923174E-42
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -1.0136233509180746E-35, -2.700523679957319E-42, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -Infinity/ division = -0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = -1.0136233509180746E-35/ division = Infinity
  numbers[j+1] = -1.0136233509180746E-35/ numbers[j+2] = -2.700523679957319E-42/ division = 3753432.559917766
  numbers[j+1] = -2.700523679957319E-42/ numbers[j+2] = 25.0/ division = -1.0802094719829276E-43
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, Infinity, 4.054493403672298E-37, -6.001163733238486E-44, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 4.054493403672298E-37/ division = Infinity
  numbers[j+1] = 4.054493403672298E-37/ numbers[j+2] = -6.001163733238486E-44/ division = -6756178.607851979
  numbers[j+1] = -6.001163733238486E-44/ numbers[j+2] = 25.0/ division = -2.4004654932953945E-45
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, Infinity, -1.6217973614689192E-38, -1.3335919407196636E-45, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -1.6217973614689192E-38/ division = -Infinity
  numbers[j+1] = -1.6217973614689192E-38/ numbers[j+2] = -1.3335919407196636E-45/ division = 1.2161121494133562E7
  numbers[j+1] = -1.3335919407196636E-45/ numbers[j+2] = 25.0/ division = -5.334367762878654E-47
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -Infinity, 6.487189445875677E-40, -2.963537646043697E-47, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -Infinity/ division = 0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = 6.487189445875677E-40/ division = -Infinity
  numbers[j+1] = 6.487189445875677E-40/ numbers[j+2] = -2.963537646043697E-47/ division = -2.189001868944041E7
  numbers[j+1] = -2.963537646043697E-47/ numbers[j+2] = 25.0/ division = -1.1854150584174788E-48
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -2.5948757783502705E-41, -6.5856392134304374E-49, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -Infinity/ division = -0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = -2.5948757783502705E-41/ division = Infinity
  numbers[j+1] = -2.5948757783502705E-41/ numbers[j+2] = -6.5856392134304374E-49/ division = 3.940203364099274E7
  numbers[j+1] = -6.5856392134304374E-49/ numbers[j+2] = 25.0/ division = -2.634255685372175E-50
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, Infinity, 1.0379503113401082E-42, -1.4634753807623195E-50, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 1.0379503113401082E-42/ division = Infinity
  numbers[j+1] = 1.0379503113401082E-42/ numbers[j+2] = -1.4634753807623195E-50/ division = -7.092366055378693E7
  numbers[j+1] = -1.4634753807623195E-50/ numbers[j+2] = 25.0/ division = -5.853901523049278E-52
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, Infinity, -4.151801245360433E-44, -3.2521675128051545E-52, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -4.151801245360433E-44/ division = -Infinity
  numbers[j+1] = -4.151801245360433E-44/ numbers[j+2] = -3.2521675128051545E-52/ division = 1.2766258899681647E8
  numbers[j+1] = -3.2521675128051545E-52/ numbers[j+2] = 25.0/ division = -1.3008670051220617E-53
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -Infinity, 1.6607204981441733E-45, -7.227038917344787E-54, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -Infinity/ division = 0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = 1.6607204981441733E-45/ division = -Infinity
  numbers[j+1] = 1.6607204981441733E-45/ numbers[j+2] = -7.227038917344787E-54/ division = -2.297926601942697E8
  numbers[j+1] = -7.227038917344787E-54/ numbers[j+2] = 25.0/ division = -2.890815566937915E-55
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -6.642881992576694E-47, -1.6060086482988416E-55, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -Infinity/ division = -0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = -6.642881992576694E-47/ division = Infinity
  numbers[j+1] = -6.642881992576694E-47/ numbers[j+2] = -1.6060086482988416E-55/ division = 4.1362678834968543E8
  numbers[j+1] = -1.6060086482988416E-55/ numbers[j+2] = 25.0/ division = -6.424034593195367E-57
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, Infinity, 2.6571527970306775E-48, -3.568908107330759E-57, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 2.6571527970306775E-48/ division = Infinity
  numbers[j+1] = 2.6571527970306775E-48/ numbers[j+2] = -3.568908107330759E-57/ division = -7.445282190294338E8
  numbers[j+1] = -3.568908107330759E-57/ numbers[j+2] = 25.0/ division = -1.4275632429323038E-58
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, Infinity, -1.062861118812271E-49, -7.930906905179465E-59, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -1.062861118812271E-49/ division = -Infinity
  numbers[j+1] = -1.062861118812271E-49/ numbers[j+2] = -7.930906905179465E-59/ division = 1.340150794252981E9
  numbers[j+1] = -7.930906905179465E-59/ numbers[j+2] = 25.0/ division = -3.172362762071786E-60
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -Infinity, 4.2514444752490843E-51, -1.7624237567065477E-60, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -Infinity/ division = 0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = 4.2514444752490843E-51/ division = -Infinity
  numbers[j+1] = 4.2514444752490843E-51/ numbers[j+2] = -1.7624237567065477E-60/ division = -2.4122714296553655E9
  numbers[j+1] = -1.7624237567065477E-60/ numbers[j+2] = 25.0/ division = -7.049695026826191E-62
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -1.7005777900996337E-52, -3.916497237125661E-62, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -Infinity/ division = -0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = -1.7005777900996337E-52/ division = Infinity
  numbers[j+1] = -1.7005777900996337E-52/ numbers[j+2] = -3.916497237125661E-62/ division = 4.342088573379659E9
  numbers[j+1] = -3.916497237125661E-62/ numbers[j+2] = 25.0/ division = -1.5665988948502644E-63
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, Infinity, 6.802311160398535E-54, -8.70332719361258E-64, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 6.802311160398535E-54/ division = Infinity
  numbers[j+1] = 6.802311160398535E-54/ numbers[j+2] = -8.70332719361258E-64/ division = -7.815759432083386E9
  numbers[j+1] = -8.70332719361258E-64/ numbers[j+2] = 25.0/ division = -3.481330877445032E-65
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, Infinity, -2.7209244641594143E-55, -1.9340727096916842E-65, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -2.7209244641594143E-55/ division = -Infinity
  numbers[j+1] = -2.7209244641594143E-55/ numbers[j+2] = -1.9340727096916842E-65/ division = 1.40683669777501E10
  numbers[j+1] = -1.9340727096916842E-65/ numbers[j+2] = 25.0/ division = -7.736290838766737E-67
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -Infinity, 1.0883697856637656E-56, -4.297939354870409E-67, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -Infinity/ division = 0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = 1.0883697856637656E-56/ division = -Infinity
  numbers[j+1] = 1.0883697856637656E-56/ numbers[j+2] = -4.297939354870409E-67/ division = -2.5323060559950176E10
  numbers[j+1] = -4.297939354870409E-67/ numbers[j+2] = 25.0/ division = -1.7191757419481636E-68
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -4.353479142655063E-58, -9.550976344156465E-69, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -Infinity/ division = -0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = -4.353479142655063E-58/ division = Infinity
  numbers[j+1] = -4.353479142655063E-58/ numbers[j+2] = -9.550976344156465E-69/ division = 4.558150900791032E10
  numbers[j+1] = -9.550976344156465E-69/ numbers[j+2] = 25.0/ division = -3.820390537662586E-70
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, Infinity, 1.741391657062025E-59, -2.1224391875903257E-70, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 1.741391657062025E-59/ division = Infinity
  numbers[j+1] = 1.741391657062025E-59/ numbers[j+2] = -2.1224391875903257E-70/ division = -8.204671621423857E10
  numbers[j+1] = -2.1224391875903257E-70/ numbers[j+2] = 25.0/ division = -8.489756750361302E-72
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, Infinity, -6.9655666282481E-61, -4.716531527978501E-72, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -6.9655666282481E-61/ division = -Infinity
  numbers[j+1] = -6.9655666282481E-61/ numbers[j+2] = -4.716531527978501E-72/ division = 1.4768408918562943E11
  numbers[j+1] = -4.716531527978501E-72/ numbers[j+2] = 25.0/ division = -1.8866126111914005E-73
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -Infinity, 2.78622665129924E-62, -1.048118117328556E-73, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -Infinity/ division = 0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = 2.78622665129924E-62/ division = -Infinity
  numbers[j+1] = 2.78622665129924E-62/ numbers[j+2] = -1.048118117328556E-73/ division = -2.658313605341329E11
  numbers[j+1] = -1.048118117328556E-73/ numbers[j+2] = 25.0/ division = -4.192472469314224E-75
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -1.1144906605196959E-63, -2.3291513718412356E-75, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = 0.0/ division = NaN
  numbers[j+1] = 0.0/ numbers[j+2] = -Infinity/ division = -0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = -1.1144906605196959E-63/ division = Infinity
  numbers[j+1] = -1.1144906605196959E-63/ numbers[j+2] = -2.3291513718412356E-75/ division = 4.784964489614392E11
  numbers[j+1] = -2.3291513718412356E-75/ numbers[j+2] = 25.0/ division = -9.316605487364942E-77
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, Infinity, 4.457962642078784E-65, -5.175891937424968E-77, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 4.457962642078784E-65/ division = Infinity
  numbers[j+1] = 4.457962642078784E-65/ numbers[j+2] = -5.175891937424968E-77/ division = -8.612936081305906E11
  numbers[j+1] = -5.175891937424968E-77/ numbers[j+2] = 25.0/ division = -2.070356774969987E-78
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[-0.0, -0.0, Infinity, -0.0, -0.0, Infinity, -1.7831850568315134E-66, -1.1501982083166594E-78, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -0.0/ division = -Infinity
  numbers[j+1] = -0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = -1.7831850568315134E-66/ division = -Infinity
  numbers[j+1] = -1.7831850568315134E-66/ numbers[j+2] = -1.1501982083166594E-78/ division = 1.5503284946350632E12
  numbers[j+1] = -1.1501982083166594E-78/ numbers[j+2] = 25.0/ division = -4.600792833266638E-80
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
[0.0, -0.0, Infinity, 0.0, -0.0, -Infinity, 7.132740227326054E-68, -2.5559960184814653E-80, 25.0, -20.0]
  numbers[j+1] = -0.0/ numbers[j+2] = Infinity/ division = -0.0
  numbers[j+1] = Infinity/ numbers[j+2] = 0.0/ division = Infinity
  numbers[j+1] = 0.0/ numbers[j+2] = -0.0/ division = NaN
  numbers[j+1] = -0.0/ numbers[j+2] = -Infinity/ division = 0.0
  numbers[j+1] = -Infinity/ numbers[j+2] = 7.132740227326054E-68/ division = -Infinity
  numbers[j+1] = 7.132740227326054E-68/ numbers[j+2] = -2.5559960184814653E-80/ division = -2.790591290343114E12
  numbers[j+1] = -2.5559960184814653E-80/ numbers[j+2] = 25.0/ division = -1.0223984073925861E-81
  numbers[j+1] = 25.0/ numbers[j+2] = -20.0/ division = -1.25
Final list [-0.0, -0.0, Infinity, 0.0, 0.0, -Infinity, -2.8530960909304215E-69, -5.679991152181034E-82, 25.0, -20.0]
Remarks

This output clearly shows us two things:

  • The error comes from a subtraction -0.0 - (-0.0) = 0.0, which sets the denominator of the division to 0.
  • Apparition of these 0s is caused by a very fast decrease of all values in the list (except the last two).

This output shows us two things:

  • The error comes from a division 3.5381129532088862E277 / 2.392123271624348E-165 = Infinity, which sets the denominator of the division to Infinity, which in turn introduces 0s.
  • Apparition of these values is caused by a very fast decrease of all values in the list (except the last two).

We have found the source of our problem! Correcting it obviously depends on what the algorithm was supposed to do, as this is a toy example.

Still note that the error is not caused by the presence of two consecutive random numbers in the initial list, which could have been the easiest solution you may have thought of! Since errors are not always obvious, always execute your programs frequently, and use debugging tools!

Now, use the step-by-step debugger of VSCode to debug the same program. How do you do it?

Correction

As the error occurs on line 14, let’s put a breakpoint at that line. Then, running the step-by-step debugger, let’s see the contents of variables everytime it reaches this breakpoint:

i
j
numbers
0 0 [44, -38, 27, -10, -17, -28, -17, 19, -19, 45]
0 1 [-0.676923076923077, -38, 27, -10, -17, -28, -17, 19, -19, 45]
0 2 [-0.676923076923077, -1.027027027027027, 27, -10, -17, -28, -17, 19, -19, 45]
… … …
14 2 [-0.0, -0.0, -inf, -0.0, 0.0, -3.238439139240665e+104,
-1.080447209418469e-17, 9.822769774067204e-25, -19, 45]

3 — Step-by-step programming

You are going to implement a Monte Carlo strategy to estimate the value of $\pi$. The principle illustrated in the figure below is to generate pseudo-randomly the Cartesian coordinates of points located in the unit subspace, the dark cyan square. Considering $n$ pseudo-randomly generated points, the ratio between the number of points located in the top right-hand quarter of the disc and the total number of points generated is an estimate of $\frac{\pi}{4}$.

To determine if a point falls in the top right-hand quarter of the disc, one just has to compute its Euclidean distance from the origin. If this distance is less than 1, it is in the disc. As a recall, for a point whose coordinates are $(x,y)$, the distance from origin is computed as follows: $\sqrt{x^2 + y^2}$.

3.1 — Get your tools

To estimate $\pi$, you need a few programming tools:

  • A random number generator, and a method of that generator to obtain a number in the $[0, 1]$ interval. Find the correct method in the documentation of the random module.

  • A method to compute the square root of a value. Find the correct method in the documentation of the math module.

You will ask through a prompt the number of points to generate (the value to assign to $n$). To do so, use the input built-in function.

Let’s follow some steps to reach a functioning program.

3.2 — Split your algorithm into pieces

A complete source code is always written step-by-step, and each step tested. It is clearly the best and only way to efficiency reach the objective. So, in your opinion, what should we do?

--- secondary_color: lightgray --- # Quiz Put the proposed steps in order! 1. Ask for the value of $n$. 2. Define a loop that will make $n$ turns. 3. Generate random points in the unit subspace. 4. Compute the distance between a pseudo-random point and the origin. 5. Estimate the value of $\pi$.

Implement this algorithm one step after the other, and use the print statement to check that every step seems correct.

Correction

Let’s write the program one step after the other. First, let’s ask for a value of $n$ and check that the input works as expected. We also directly add the imports and the random seed. The print helps us check that the input works well, and that our type conversion is fine.

​
# Number of points to generate
n = int(input("Enter the number of points to generate: "))
print(n, type(n))
// Needed imports
import java.util.Scanner;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Create a scanner object for user input
        Scanner scanner = new Scanner(System.in);

        // Number of points to generate
        System.out.print("Enter the number of points to generate: ");
        int n = scanner.nextInt();

        // Print the value of n and its type
        System.out.println(n + " " + ((Object)n).getClass().getSimpleName());
    }

}

Now, let’s generate $n$ points. Here, we choose to set the random seed for easier debugging just in case we need it.

​
# Needed imports
import random

# Set the seed
random.seed(42)

# Number of points to generate
n = int(input("Enter the number of points to generate: "))

# Generate the points
for i in range(n):
    x = random.random()
    y = random.random()
    print("(%f, %f)" % (x, y))
// Needed imports
import java.util.Scanner;
import java.util.Random;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Create a scanner object for user input
        Scanner scanner = new Scanner(System.in);

        // Set the seed
        Random random = new Random();
        random.setSeed(42);

        // Number of points to generate
        System.out.print("Enter the number of points to generate: ");
        int n = scanner.nextInt();

        // Generate the points
        for (int i = 0; i < n; i++) {
            double x = random.nextDouble();
            double y = random.nextDouble();
            System.out.printf("(%.6f, %.6f)\n", x, y);
        }
    }

}

Everything seems right, so let’s continue. To estimate $\pi$, determine the number of points that fall in the unit circle (computation of the norm from the origin to the point).

​
# Needed imports
import random
from math import sqrt

# Set the seed
random.seed(42)

# Number of points to generate
n = int(input("Enter the number of points to generate: "))

# Number of points inside the circle (initially 0)
nb_pt_inside = 0
for i in range(n):

    # Generate the point
    x = random.random()
    y = random.random()
    
    # Check if the point is inside the circle
    if sqrt(x**2 + y**2) <= 1:
        nb_pt_inside += 1

# Estimate pi
estimate_pi = 4 * nb_pt_inside / n
print("Estimate of pi: ", estimate_pi)
// Needed imports
import java.util.Scanner;
import java.util.Random;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {
  
    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Create a scanner object for user input
        Scanner scanner = new Scanner(System.in);

        // Set the seed
        Random random = new Random();
        random.setSeed(42);

        // Number of points to generate
        System.out.print("Enter the number of points to generate: ");
        int n = scanner.nextInt();

        // Number of points inside the circle (initially 0)
        int nbPtInside = 0;
        for (int i = 0; i < n; i++) {

            // Generate the point
            double x = random.nextDouble();
            double y = random.nextDouble();

            // Check if the point is inside the circle
            if (Math.sqrt(x * x + y * y) <= 1) {
                nbPtInside++;
            }
        }

        // Estimate pi
        double estimatePi = 4.0 * nbPtInside / n;
        System.out.println("Estimate of pi: " + estimatePi);
    }

}

Varying the value of $n$, observe the precision of the estimate of $\pi$ obtained.

Correction

Here are some possible results we observed. Note that because we use random numbers, you may get different results.

N
Estimate of pi
10 3.6
100 3.12
1,000 3.128
10,000 3.126
100,000 3.13728
1,000,000 3.140592
10,000,000 3.1423096
100,000,000 3.14147488

4 — A classical sorting problem

Sorting an array (or a list) of values according to an ordering relation is a very classical algorithmic problem. It is often used to empirically discover theoretical concepts such as termination, complexity, etc.

4.1 — Two variants of the selection sort strategy

The selection sort algorithm is very easy to understand (here is an explanation with music and gesture). Consider you have a set of unsorted cards in your right hand and that you progressively construct a sorted permutation of these cards in your left hand. The selection sort strategy is to first select the card with the lowest value in your right hand and to place it in the first position in your left hand. You then do the same with the card having the second lowest value, and so on.

Identify the different steps to finally have a program that generates an array of size N (a chosen constant) composed of randomly generated values, then sorts by selection this array and checks that the array is indeed sorted. Then write a program that performs that sort.

Once the program is ready, write an additional step to verify that the array is indeed sorted. Test its correctness by trying different values for N (both odd and even). Additionally, ensure the program handles multiple occurrences of the same values in the array.

Correction
​
# Needed imports
import random
import time

# Constants
N = 10

# Set the seed
random.seed(42)

# Generate a random list of N numbers
list_to_sort = [random.randint(0, 100) for i in range(N)]
print("Unsorted list:", list_to_sort)

# Get the current time
# We are going to measure execution time for info
# Here, we measure the process time rather than the elapsed time
start = time.process_time()

# Iterate over each cell of the list
for i in range(N-1):

    # Find the smallest value in the remaining part of the list (i.e., from i to N-1)
    # Below, we provide an implementation based on loops
    # It is equivalent to the following code:
    # index_min = numpy.argmin(list_to_sort[:i])
    index_min = i
    for j in range(i+1, N):
        if list_to_sort[j] < list_to_sort[index_min]:
            index_min = j

    # Exchange value at position i with the smallest value found
    list_to_sort[i], list_to_sort[index_min] = list_to_sort[index_min], list_to_sort[i]

# Get the current time after sorting
# The difference will give us the execution time
stop = time.process_time()
print("Sorted list in %f seconds" % (stop - start))

# Check that the list is indeed sorted
i = 1
while i < N and list_to_sort[i] >= list_to_sort[i-1]:
    i += 1
print("Is the list sorted?", i == N)
print("Sorted list:", list_to_sort)
// Needed imports
import java.util.Random;
import java.util.Arrays;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    // Constants
    public static final int N = 10;

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Create a random object
        Random random = new Random();
        random.setSeed(42);

        // Generate a random list of N numbers
        int[] listToSort = new int[N];
        for (int i = 0; i < N; i++) {
            listToSort[i] = random.nextInt(101);
        }
        System.out.println("Unsorted list: " + Arrays.toString(listToSort));

        // Get the current process time (in nanoseconds)
        long start = System.nanoTime();

        // Selection sort implementation
        for (int i = 0; i < N - 1; i++) {
            // Find the smallest value in the remaining part of the list (i.e., from i to N-1)
            int indexMin = i;
            for (int j = i + 1; j < N; j++) {
                if (listToSort[j] < listToSort[indexMin]) {
                    indexMin = j;
                }
            }

            // Swap value at position i with the smallest value found
            int temp = listToSort[i];
            listToSort[i] = listToSort[indexMin];
            listToSort[indexMin] = temp;
        }

        // Get the current time after sorting (in nanoseconds)
        long stop = System.nanoTime();
        System.out.printf("Sorted list in %.6f seconds%n", (stop - start) / 1e9);

        // Check if the list is sorted
        boolean isSorted = true;
        for (int i = 1; i < N; i++) {
            if (listToSort[i] < listToSort[i - 1]) {
                isSorted = false;
                break;
            }
        }
        System.out.println("Is the list sorted? " + isSorted);
        System.out.println("Sorted list: " + Arrays.toString(listToSort));
    }

}

A slight variation of the algorithm consists in positioning the lowest and largest value at the same time, the second lowest and the second largest, and so on. Implement this strategy in a new script.

Correction
​
# Needed imports
import random
import time

# Constants
N = 10

# Set the seed
random.seed(42)

# Generate a random list of N numbers
list_to_sort = [random.randint(0, 100) for i in range(N)]
print("Unsorted list:", list_to_sort)

# Get the current time
# We are going to measure execution time for info
# Here, we measure the process time rather than the elapsed time
start = time.process_time()

# Iterate over each cell of the list
for i in range((N-1) // 2 + 1):
    
    # Find the smallest and the highest value in the remaining part of the list (i.e., from i to N-1)
    # Below, we provide an implementation based on loops
    # It is equivalent to the following code:
    # index_min = numpy.argmin(list_to_sort[:i])
    # index_max = numpy.argmax(list_to_sort[:i])
    index_min = i
    index_max = i
    for j in range(i+1, N-i):
        if list_to_sort[j] < list_to_sort[index_min]:
            index_min = j
        if list_to_sort[j] > list_to_sort[index_max]:
            index_max = j

    # Exchange value at position i with the smallest value found
    list_to_sort[i], list_to_sort[index_min] = list_to_sort[index_min], list_to_sort[i]

    # In case the highest value is located at position i and would be erased by the lowest value found
    if index_max == i:
        index_max = index_min

    # Exchange value at position N-i-1 with the highest value found
    list_to_sort[N-i-1], list_to_sort[index_max] = list_to_sort[index_max], list_to_sort[N-i-1]

# Get the current time after sorting
# The difference will give us the execution time
stop = time.process_time()
print("Sorted list in %f seconds" % (stop - start))

# Check that the list is indeed sorted
i = 1
while i < N and list_to_sort[i] >= list_to_sort[i-1]:
    i += 1
print("Is the list sorted?", i == N)
print("Sorted list:", list_to_sort)
// Needed imports
import java.util.Random;
import java.util.Arrays;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    // Constants
    public static final int N = 10;

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Create a random object
        Random random = new Random();
        random.setSeed(42);

        // Generate a random list of N numbers
        int[] listToSort = new int[N];
        for (int i = 0; i < N; i++) {
            listToSort[i] = random.nextInt(101);
        }
        System.out.println("Unsorted list: " + Arrays.toString(listToSort));

        // Get the current process time (in nanoseconds)
        long start = System.nanoTime();

        // Iterate over each cell of the list
        for (int i = 0; i <= (N - 1) / 2; i++) {
            // Find the smallest and the largest value in the remaining part of the list (i.e., from i to N-1)
            int indexMin = i;
            int indexMax = i;
            for (int j = i + 1; j < N - i; j++) {
                if (listToSort[j] < listToSort[indexMin]) {
                    indexMin = j;
                }
                if (listToSort[j] > listToSort[indexMax]) {
                    indexMax = j;
                }
            }

            // Swap value at position i with the smallest value found
            int temp = listToSort[i];
            listToSort[i] = listToSort[indexMin];
            listToSort[indexMin] = temp;

            // If the highest value is located at position i and would be erased by the smallest value found
            if (indexMax == i) {
                indexMax = indexMin;
            }

            // Swap value at position N-i-1 with the largest value found
            temp = listToSort[N - i - 1];
            listToSort[N - i - 1] = listToSort[indexMax];
            listToSort[indexMax] = temp;
        }

        // Get the current time after sorting (in nanoseconds)
        long stop = System.nanoTime();
        System.out.printf("Sorted list in %.6f seconds%n", (stop - start) / 1e9);

        // Check if the list is sorted
        boolean isSorted = true;
        for (int i = 1; i < N; i++) {
            if (listToSort[i] < listToSort[i - 1]) {
                isSorted = false;
                break;
            }
        }
        System.out.println("Is the list sorted? " + isSorted);
        System.out.println("Sorted list: " + Arrays.toString(listToSort));
    }

}

4.2 — Empirical comparison of their efficiency

For each of the sorting strategies you developed in the previous section, make sure that you create a dedicated script. In the following, we assume that the first one is called selection_sort_min.py, and the second selection_sort_min_max.py.

Then, adapt both scripts as follows:

  • The only print they make should be the elapsed time for sorting the list, i.e., print(stop - start) in the corrections above (without any other text). Remove all other prints, and the final test to verify is the list is sorted.

  • Adapt your script to retrieve the value of N from the command line first argument, i.e., using sys.argv[1]. Do not forget to convert it to an integer.

  • Adapt your script to retrieve the value of the random seed from the command line second argument, i.e., using sys.argv[2]. Do not forget to convert it to an integer.

Correction
​
# Needed imports
import random
import time
import sys

# Constants
N = int(sys.argv[1])

# Set the seed
random.seed(int(sys.argv[2]))

# Generate a random list of N numbers
list_to_sort = [random.randint(0, 100) for i in range(N)]

# Get the current time
# We are going to measure execution time for info
# Here, we measure the process time rather than the elapsed time
start = time.process_time()

# Iterate over each cell of the list
for i in range(N-1):

    # Find the smallest value in the remaining part of the list (i.e., from i to N-1)
    # Below, we provide an implementation based on loops
    # It is equivalent to the following code:
    # index_min = numpy.argmin(list_to_sort[:i])
    index_min = i
    for j in range(i+1, N):
        if list_to_sort[j] < list_to_sort[index_min]:
            index_min = j

    # Exchange value at position i with the smallest value found
    list_to_sort[i], list_to_sort[index_min] = list_to_sort[index_min], list_to_sort[i]

# Get the current time after sorting
# The difference will give us the execution time
stop = time.process_time()
print(stop - start)
// Needed imports
import java.util.Random;

/**
 * To run this code, you need to have Java installed on your computer, then:
  * - Create a file named `Main.java` in a directory of your choice.
  * - Copy this code in the file.
  * - Open a terminal in the directory where the file is located.
  * - Run the command `javac Main.java` to compile the code.
  * - Run the command `java -ea Main` to execute the compiled code.
  * Note: '-ea' is an option to enable assertions in Java.
  */
public class Main {

    /**
     * This is the entry point of your program.
      * It contains the first codes that are going to be executed.
      *
      * @param args Command line arguments received.
      */
    public static void main(String[] args) {
        // Constants: Read N and seed from command line arguments
        int N = Integer.parseInt(args[0]);
        long seed = Long.parseLong(args[1]);

        // Create a random object and set the seed
        Random random = new Random(seed);

        // Generate a random list of N numbers
        int[] listToSort = new int[N];
        for (int i = 0; i < N; i++) {
            listToSort[i] = random.nextInt(101);  // Generate random integers between 0 and 100
        }

        // Get the current process time (in nanoseconds)
        long start = System.nanoTime();

        // Selection sort algorithm
        for (int i = 0; i < N - 1; i++) {
            // Find the smallest value in the remaining part of the list (i.e., from i to N-1)
            int indexMin = i;
            for (int j = i + 1; j < N; j++) {
                if (listToSort[j] < listToSort[indexMin]) {
                    indexMin = j;
                }
            }

            // Swap value at position i with the smallest value found
            int temp = listToSort[i];
            listToSort[i] = listToSort[indexMin];
            listToSort[indexMin] = temp;
        }

        // Get the current time after sorting (in nanoseconds)
        long stop = System.nanoTime();
        
        // Print the execution time in seconds
        System.out.println((stop - start) / 1e9);
    }

}
​
# Needed imports
import random
import time
import sys

# Constants
N = int(sys.argv[1])

# Set the seed
random.seed(int(sys.argv[2]))

# Generate a random list of N numbers
list_to_sort = [random.randint(0, 100) for i in range(N)]

# Get the current time
# We are going to measure execution time for info
# Here, we measure the process time rather than the elapsed time
start = time.process_time()

# Iterate over each cell of the list
for i in range((N-1) // 2 + 1):
    
    # Find the smallest and the highest value in the remaining part of the list (i.e., from i to N-1)
    # Below, we provide an implementation based on loops
    # It is equivalent to the following code:
    # index_min = numpy.argmin(list_to_sort[:i])
    # index_max = numpy.argmax(list_to_sort[:i])
    index_min = i
    index_max = i
    for j in range(i+1, N-i):
        if list_to_sort[j] < list_to_sort[index_min]:
            index_min = j
        if list_to_sort[j] > list_to_sort[index_max]:
            index_max = j

    # Exchange value at position i with the smallest value found
    list_to_sort[i], list_to_sort[index_min] = list_to_sort[index_min], list_to_sort[i]

    # In case the highest value is located at position i and would be erased by the lowest value found
    if index_max == i:
        index_max = index_min

    # Exchange value at position N-i-1 with the highest value found
    list_to_sort[N-i-1], list_to_sort[index_max] = list_to_sort[index_max], list_to_sort[N-i-1]

# Get the current time after sorting
# The difference will give us the execution time
stop = time.process_time()
print(stop - start)
// Needed imports
import java.util.Random;

/**
 * To run this code, you need to have Java installed on your computer, then:
  * - Create a file named `Main.java` in a directory of your choice.
  * - Copy this code in the file.
  * - Open a terminal in the directory where the file is located.
  * - Run the command `javac Main.java` to compile the code.
  * - Run the command `java -ea Main` to execute the compiled code.
  * Note: '-ea' is an option to enable assertions in Java.
  */
public class Main {

    /**
     * This is the entry point of your program.
      * It contains the first codes that are going to be executed.
      *
      * @param args Command line arguments received.
      */
    public static void main(String[] args) {
        // Constants: Read N and seed from command line arguments
        int N = Integer.parseInt(args[0]);
        long seed = Long.parseLong(args[1]);

        // Create a random object and set the seed
        Random random = new Random(seed);

        // Generate a random list of N numbers
        int[] listToSort = new int[N];
        for (int i = 0; i < N; i++) {
            listToSort[i] = random.nextInt(101);  // Generate random integers between 0 and 100
        }

        // Get the current process time (in nanoseconds)
        long start = System.nanoTime();

        // Iterate over each cell of the list
        for (int i = 0; i <= (N - 1) / 2; i++) {
            // Find the smallest and the largest value in the remaining part of the list (i.e., from i to N-1)
            int indexMin = i;
            int indexMax = i;
            for (int j = i + 1; j < N - i; j++) {
                if (listToSort[j] < listToSort[indexMin]) {
                    indexMin = j;
                }
                if (listToSort[j] > listToSort[indexMax]) {
                    indexMax = j;
                }
            }

            // Swap value at position i with the smallest value found
            int temp = listToSort[i];
            listToSort[i] = listToSort[indexMin];
            listToSort[indexMin] = temp;

            // If the highest value is located at position i and would be erased by the smallest value found
            if (indexMax == i) {
                indexMax = indexMin;
            }

            // Swap value at position N-i-1 with the largest value found
            temp = listToSort[N - i - 1];
            listToSort[N - i - 1] = listToSort[indexMax];
            listToSort[indexMax] = temp;
        }

        // Get the current time after sorting (in nanoseconds)
        long stop = System.nanoTime();

        // Print the execution time in seconds
        System.out.println((stop - start) / 1e9);
    }

}

Now, create a new Python script that uses the built-in sorted Python function. Below, we assume that this script is named built_in_sorted.py. It should respect the same adaptations as the other two scripts above, i.e.,

  • Be runnable from the command line using the following command:

    ​
    python <script_name> <N> <seed>
    python <script_name> <N> <seed>
    python3 <script_name> <N> <seed>
    python3 <script_name> <N> <seed>

  • Print the total elapsed time as a result.

Correction
​
# Needed imports
import random
import time
import sys

# Constants
N = int(sys.argv[1])

# Set the seed
random.seed(int(sys.argv[2]))

# Generate a random list of N numbers
list_to_sort = [random.randint(0, 100) for i in range(N)]

# Get the current time
# We are going to measure execution time for info
# Here, we measure the process time rather than the elapsed time
start = time.process_time()

# Sort the list
list_to_sort = sorted(list_to_sort)

# Get the current time after sorting
# The difference will give us the execution time
stop = time.process_time()
print(stop - start)
// Needed imports
import java.util.Random;
import java.util.Arrays;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Constants: Read N and seed from command line arguments
        int N = Integer.parseInt(args[0]);
        long seed = Long.parseLong(args[1]);

        // Create a random object and set the seed
        Random random = new Random(seed);

        // Generate a random list of N numbers
        int[] listToSort = new int[N];
        for (int i = 0; i < N; i++) {
            listToSort[i] = random.nextInt(101);  // Generate random integers between 0 and 100
        }

        // Get the current process time (in nanoseconds)
        long start = System.nanoTime();

        // Sort the list using Java's built-in sorting method
        Arrays.sort(listToSort);

        // Get the current time after sorting
        long stop = System.nanoTime();

        // Print the execution time in seconds
        System.out.println((stop - start) / 1e9);
    }

}

This function uses a more advanced algorithm called Powersort. Also, when calling sorted, the interpreter calls a compiled version of the sorting function in the background. Therefore, it can directly be interpreted as binary code by your computer, and is therefore a lot faster.

Important

Most of Python built-in functions (and those of libraries) use the same trick, so it is always better to use them rather than reinvent the wheel. The fewer loops you write, the faster your Python code!

Now, put the following script in the same directory as the files you just created. Don’t forget to adapt file names This script will run all other scripts for various input sizes, and create a .csv file with the average times. It will also use the random seed to make sure that all scripts are compared in the same settings.

​
# Needed imports
import subprocess
import os

# Constants
SCRIPTS = ['selection_sort_min.py', 'selection_sort_min_max.py', 'built_in_sorted.py']
NB_TESTS_FOR_AVERAGE = 10
VALUES_OF_N = [10, 100, 1000, 10000]
OUTPUT_FILE_NAME = 'average_execution_times.csv'
PYTHON_EXECUTABLE = "python"

# We will write the average execution time for each script in a file
current_dir = os.path.dirname(__file__)
output_file = open(os.path.join(current_dir, OUTPUT_FILE_NAME), 'w', buffering=1)

# Loop over each script
for script in SCRIPTS:

    # Loop over each value of N
    results_for_n = []
    for n in VALUES_OF_N:

        # Loop over each test
        execution_times = []
        for seed in range(NB_TESTS_FOR_AVERAGE):

            # Run the script
            command = [PYTHON_EXECUTABLE, os.path.join(current_dir, script), str(n), str(seed)]
            output = subprocess.run(command, capture_output=True, text=True)
            result = float(output.stdout)
            execution_times.append(result)

        # Compute the average execution time
        average_execution_time = sum(execution_times) / NB_TESTS_FOR_AVERAGE
        results_for_n.append(average_execution_time)

    # Write the results for the current script
    output_file.write(script + ';')
    output_file.write(';'.join([str(result) for result in results_for_n]))
    output_file.write('\n')

# Close the file
output_file.close()

Then, use either LibreOffice, Microsoft Office, or any equivalent software, to plot, for each script, the average time as a function of list size. Alternately, you can also install the matplotlib Python package to create this curve. Here is a script to do it.

​
# Needed imports
import os
import matplotlib.pyplot as plt

# Constants
VALUES_OF_N = [10, 100, 1000, 10000]
CSV_FILE_NAME = 'average_execution_times.csv'

# Read the results from the file
current_dir = os.path.dirname(__file__)
results = {}
with open(os.path.join(current_dir, CSV_FILE_NAME), 'r') as file:
    for line in file:
        split_line = line.strip().split(';')
        script = split_line[0]
        times = [float(time) for time in split_line[1:]]
        results[script] = times

# Plot the results
fig, ax = plt.subplots()
for script, times in results.items():
    ax.plot(VALUES_OF_N, times, label=script)
ax.set_xscale("log")
ax.set_yscale("log")
ax.set_xlabel("N")
ax.set_ylabel("Average execution time (s)")
ax.set_title("Comparison of sorting algorithms")
ax.legend()
plt.show()
Correction

To go further

5 — A classical sorting problem (again)

Design an algorithm based on sorting comparisons that satisfies the following invariant:

After i iterations of the inner loop, the i highest values of the list l of size N are sorted in an ascending order in the right side of l.

At each iteration, the first value of the array (index 0) is moved by swapping neighbours to the right until a highest value is found. In this case, it is this highest value that is moved to the right side of the array. An inner loop is used to apply the previous process until no more swaps can be performed anymore.

Write your script in a new file called bubble_sort.py. Make sure that your algorithm works correctly, as you did in a previous exercise.

Correction

The correction, with music and images!

​
# Needed imports
import random
import time

# Constants
N = 10

# Set the seed
random.seed(42)

# Generate a random list of N numbers
list_to_sort = [random.randint(0, 100) for i in range(N)]
print("Unsorted list:", list_to_sort)

# Get the current time
# We are going to measure execution time for info
# Here, we measure the process time rather than the elapsed time
start = time.process_time()

# Loop through the list to sort it
for i in range(len(list_to_sort)):

    # Loop through the list to compare elements
    for j in range(len(list_to_sort) - 1):

        # Swap elements if they are not in the right order
        if list_to_sort[j] > list_to_sort[j + 1]:
            list_to_sort[j], list_to_sort[j + 1] = list_to_sort[j + 1], list_to_sort[j]

# Get the current time after sorting
# The difference will give us the execution time
stop = time.process_time()
print("Sorted list in %f seconds" % (stop - start))

# Check that the list is indeed sorted
i = 1
while i < N and list_to_sort[i] >= list_to_sort[i-1]:
    i += 1
print("Is the list sorted?", i == N)
print("Sorted list:", list_to_sort)  
// Needed imports
import java.util.Random;
import java.util.Arrays;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    // Constants
    public static final int N = 10;

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Set the seed
        Random random = new Random(42);

        // Generate a random list of N numbers
        int[] listToSort = new int[N];
        for (int i = 0; i < N; i++) {
            listToSort[i] = random.nextInt(101);  // Generate random integers between 0 and 100
        }
        System.out.println("Unsorted list: " + Arrays.toString(listToSort));

        // Get the current process time (in nanoseconds)
        long start = System.nanoTime();

        // Bubble sort implementation
        for (int i = 0; i < listToSort.length; i++) {
            for (int j = 0; j < listToSort.length - 1; j++) {
                if (listToSort[j] > listToSort[j + 1]) {
                    // Swap elements
                    int temp = listToSort[j];
                    listToSort[j] = listToSort[j + 1];
                    listToSort[j + 1] = temp;
                }
            }
        }

        // Get the current time after sorting (in nanoseconds)
        long stop = System.nanoTime();

        // Print the execution time in seconds
        System.out.printf("Sorted list in %.6f seconds%n", (stop - start) / 1e9);

        // Check if the list is sorted
        boolean isSorted = true;
        for (int i = 1; i < N; i++) {
            if (listToSort[i] < listToSort[i - 1]) {
                isSorted = false;
                break;
            }
        }
        System.out.println("Is the list sorted? " + isSorted);
        System.out.println("Sorted list: " + Arrays.toString(listToSort));
    }
    
}

A slight improvement may certainly be applied on your algorithm to reduce the numbers of swaps. Try to find out what can be improved and compare the two versions according to the time they take to sort the same array. Write your solution in a new file called bubble_sort_2.py.

Correction

Here is a possible code that implements two optimizations:

  • Introduce a flag to detect if the list is already sorted – This allows the algorithm to stop early if no swaps are made during a pass, meaning the list is already sorted.

  • Reduce the range of the inner loop – With each pass, the largest element is bubbled to the correct position, so the range of the inner loop can be reduced with each iteration.

​
# Needed imports
import random
import time

# Constants
N = 10

# Set the seed
random.seed(42)

# Generate a random list of N numbers
list_to_sort = [random.randint(0, 100) for i in range(N)]
print("Unsorted list:", list_to_sort)

# Get the current time
# We are going to measure execution time for info
# Here, we measure the process time rather than the elapsed time
start = time.process_time()

# Loop through the list to sort it
i = 0
swapped = True
while i < len(list_to_sort) and swapped:
    
    # Initialize flag to detect if any swap was made during this pass
    swapped = False  

    # Loop through the list to compare elements
    for j in range(0, N - i - 1):

        # Swap elements if they are not in the right order
        if list_to_sort[j] > list_to_sort[j + 1]:
            list_to_sort[j], list_to_sort[j + 1] = list_to_sort[j + 1], list_to_sort[j]
            swapped = True

    # Next index
    i += 1

# Get the current time after sorting
# The difference will give us the execution time
stop = time.process_time()
print("Sorted list in %f seconds" % (stop - start))

# Check that the list is indeed sorted
i = 1
while i < N and list_to_sort[i] >= list_to_sort[i-1]:
    i += 1
print("Is the list sorted?", i == N)
print("Sorted list:", list_to_sort)
// Needed imports
import java.util.Random;
import java.util.Arrays;

/**
 * To run this code, you need to have Java installed on your computer, then:
 * - Create a file named `Main.java` in a directory of your choice.
 * - Copy this code in the file.
 * - Open a terminal in the directory where the file is located.
 * - Run the command `javac Main.java` to compile the code.
 * - Run the command `java -ea Main` to execute the compiled code.
 * Note: '-ea' is an option to enable assertions in Java.
 */
public class Main {

    // Constants
    public static final int N = 10;

    /**
     * This is the entry point of your program.
     * It contains the first codes that are going to be executed.
     *
     * @param args Command line arguments received.
     */
    public static void main(String[] args) {
        // Set the seed
        Random random = new Random(42);

        // Generate a random list of N numbers
        int[] listToSort = new int[N];
        for (int i = 0; i < N; i++) {
            listToSort[i] = random.nextInt(101);  // Generate random integers between 0 and 100
        }
        System.out.println("Unsorted list: " + Arrays.toString(listToSort));

        // Get the current process time (in nanoseconds)
        long start = System.nanoTime();

        // Bubble sort implementation with early exit when no swaps occur
        int i = 0;
        boolean swapped = true;

        while (i < listToSort.length && swapped) {
            swapped = false;  // Reset the flag

            // Loop through the list to compare elements
            for (int j = 0; j < N - i - 1; j++) {
                if (listToSort[j] > listToSort[j + 1]) {
                    // Swap elements if they are not in the right order
                    int temp = listToSort[j];
                    listToSort[j] = listToSort[j + 1];
                    listToSort[j + 1] = temp;
                    swapped = true;  // Set the flag if a swap is made
                }
            }

            // Next index
            i++;
        }

        // Get the current time after sorting (in nanoseconds)
        long stop = System.nanoTime();

        // Print the execution time in seconds
        System.out.printf("Sorted list in %.6f seconds%n", (stop - start) / 1e9);

        // Check if the list is sorted
        boolean isSorted = true;
        for (int k = 1; k < N; k++) {
            if (listToSort[k] < listToSort[k - 1]) {
                isSorted = false;
                break;
            }
        }
        System.out.println("Is the list sorted? " + isSorted);
        System.out.println("Sorted list: " + Arrays.toString(listToSort));
    }

}

How can you visualize the invariant above using a step-by-step debugger?

Correction

Add a breakpoint at the end of the loop. It will allow you to visualize the sorted list at each iteration, and to check that the invariant holds.

Here is an example of visualization:

Iteration (i + 1)
list_to_sort
1 [14, 3, 81, 35, 31, 28, 17, 94, 13, 94]
2 [3, 14, 35, 31, 28, 17, 81, 13, 94, 94]
3 [3, 14, 31, 28, 17, 35, 13, 81, 94, 94]
4 [3, 14, 28, 17, 31, 13, 35, 81, 94, 94]
5 [3, 14, 17, 28, 13, 31, 35, 81, 94, 94]
6 [3, 14, 17, 13, 28, 31, 35, 81, 94, 94]
7 [3, 14, 13,17, 28, 31, 35, 81, 94, 94]
8 [3, 13, 14, 17, 28, 31, 35, 81, 94, 94]
9 [3, 3, 14, 17, 28, 31, 35, 81, 94, 94]

Note that the last iteration is not performed as the list is already sorted, which makes the invariant true for the last index as well.

To go beyond

Important

The content of this section is very optional. We suggest you directions to explore if you wish to go deeper in the current topic.

6 — Understand the script that runs the sorts

In exercise 4, we provided you a script to compare execution times of sorting algorithms. Understand this script, and adapt your bubble sort algorithms above to integrate them in the script. Then compare their execution times with other sorting algorithms you have seen earlier.

7 — Compare sorting algorithms

There are many other sorting algorithms in existence. You can for instance check this link. Try coding a few of them and compare their performance on various list sizes.

8 — Optimize your solutions

What you can now do is to use AI tools such as GitHub Copilot or ChatGPT, either to generate the solution, or to improve the first solution you came up with! Try to do this for all exercises above, to see the differences with your solutions.